Biaxial crystals indicatrix

If we try to understand the transmission of light waves in biaxial crystals, we start from the concept of the indicatrix, and to attempt to visualize what shape this must have to show the variation of refractive index with vibration direction for such crystals. From our previous knowledge of the indicatrix for uniaxial crystals, an ellipsoid of revolution with two principal refractive indices, n0 and ne, it is a simple step to see that the indicatrix for biaxial crystals will be a triaxial ellipsoid with three principal refractive indices, n7, np and na. 

For biaxial crystals, the optical indicatrix is a bilayer surface with four points of interlayer contact, which correspond to the directions of the two optical axes. In the simple case of light propagation in the principal planes XY, YZ, and XZ, the dependences of the refractive indices on the direction of light propagation are represented by a combination of an ellipse and a circle. Thus, in the principal planes, a biaxial crystal can be considered as a uniaxial crystal for example, a biaxial crystal with nz > ny > nx in the XY plane is similar to a negative uniaxial crystal with Ho=nz... 

In practice, this integral is performed on a digital computer using a measured spectrum of R versus o>, the frequency of electromagnetic radiation. The index of refraction of anisotropic solids is described by a biaxial in-dicatrix. In triclinic crystals the indicatrix is a triaxial ellipsoid formed by three principal axes of the length 2na, 2np, and 2ny which lie along the X, Y, and Z optical directions, respectively. na, np, and ny are the three principal indices of refraction (na < < ny). For absorbing crystals the... 

If the crystal is biaxial, there are three principal refractive indices (a, /3, and 7, with a less than (3 less than 7). The optical indicatrix has... 

In general the indicatrix is like a squashed (American) football. There are three principal refractive indices on perpendicular axes the maximum, the minimum, and an intermediate value perpendicular to both of these (A, C, and B in Fig. 3.13). For example, a nylon crystal has n(D parallel to chains) > n(D parallel to hydrogen-bonded sheet) > n(D parallel to intersheets). Such a material is biaxial. If two of these principal refractive indices are equal, the indicatrix is rotationally symmetric—an ellipsoid of revolution—and the material is uniaxial. For example, a polyethylene crystal has a larger refractive index for (D parallel to chains) and a smaller refractive index for any (D perpendicular to chains) (Fig. 3.14, left drawing). 

Detailed evaluation of the optical indicatrix can give useful information on the crystal symmetry. Cubic crystals and amorphous solids are optically isotropic trigonal, tetragonal, and hexagonal crystals are uniaxial orthorhombic, monoclinic, and triclinic crystals are biaxial. Extinction directions vary with wavelength in monoclinic and triclinic crystals. 

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